Kronecker’s solution of Pell’s equation for CM fields

[La solution de Kronecker des équations Pell pour les corps CM]

Masri, Riad

Annales de l'Institut Fourier, Tome 63 (2013), p. 2287-2306 / Harvested from Numdam

Résumé

Nous généralisons la solution de Kronecker des équations Pell aux corps $K$ CM dont le groupe de Galois sur $\mathbb{Q}$ est un 2-groupe abélien élémentaire. Il s’agit d’une formule qui relie les valeurs CM d’une certaine fonction modulaire de Hilbert aux produits de logarithmes des unités fondamentales. Lorsque $K$ est quadratique imaginaire, ces valeurs CM sont des nombres algébriques reliés aux unités elliptiques des corps de classes de Hilbert de $K$. Sous l’hypothèse que la conjecture de Schanuel soit vraie, nous montrons que, lorsque $K$ et de degré plus grand que 2 sur $\mathbb{Q}$, ces valeurs CM sont transcendantes.

Abstract

We generalize Kronecker’s solution of Pell’s equation to CM fields $K$ whose Galois group over $\mathbb{Q}$ is an elementary abelian 2-group. This is an identity which relates CM values of a certain Hilbert modular function to products of logarithms of fundamental units. When $K$ is imaginary quadratic, these CM values are algebraic numbers related to elliptic units in the Hilbert class field of $K$. Assuming Schanuel’s conjecture, we show that when $K$ has degree greater than 2 over $\mathbb{Q}$ these CM values are transcendental.

Détail

Publié le : 2013-01-01

MR 3237448 | Zbl 1295.11044

DOI : https://doi.org/10.5802/aif.2830
Classification:  11F41
Mots clés: point CM, fonction modulaire de Hilbert, équation Pell

Bibliographie

[1] Asai, Tetsuya On a certain function analogous to ${\rm log}_{\eta }\,(z)$, Nagoya Math. J., Tome 40 (1970), pp. 193-211 | MR 271038 | Zbl 0213.05701

[2] Bruinier, Jan Hendrik; Yang, Tonghai CM-values of Hilbert modular functions, Invent. Math., Tome 163 (2006) no. 2, pp. 229-288 | Article | MR 2207018 | Zbl 1093.11041

[3] Buell, D. A.; Williams, H. C.; Williams, K. S. On the imaginary bicyclic biquadratic fields with class-number $2$, Math. Comp., Tome 31 (1977) no. 140, pp. 1034-1042 | MR 441914 | Zbl 0379.12002

[4] Howard, Benjamin; Yang, Tonghai Intersections of Hirzebruch-Zagier divisors and CM cycles, Springer, Heidelberg, Lecture Notes in Mathematics, Tome 2041 (2012), pp. viii+140 | MR 2951750 | Zbl 1238.11069

[5] Konno, Shuji On Kronecker’s limit formula in a totally imaginary quadratic field over a totally real algebraic number field, J. Math. Soc. Japan, Tome 17 (1965), pp. 411-424 | Article | MR 183708 | Zbl 0147.03603

[6] Murty, M. Ram; Murty, V. Kumar Transcendental values of class group $L$-functions, Math. Ann., Tome 351 (2011) no. 4, pp. 835-855 | Article | MR 2854115 | Zbl 1281.11071

[7] Murty, M. Ram; Murty, V. Kumar Transcendental values of class group $L$-functions, II, Proc. Amer. Math. Soc., Tome 140 (2012) no. 9, pp. 3041-3047 | Article | MR 2917077 | Zbl 1282.11082

[8] Siegel, Carl Ludwig Lectures on advanced analytic number theory, Tata Institute of Fundamental Research, Bombay, Notes by S. Raghavan. Tata Institute of Fundamental Research Lectures on Mathematics, No. 23 (1965), pp. iii+331+iii | MR 262150 | Zbl 0278.10001

[9] Waldschmidt, Michel Diophantine approximation on linear algebraic groups, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 326 (2000), pp. xxiv+633 (Transcendence properties of the exponential function in several variables) | MR 1756786 | Zbl 0944.11024

[10] Washington, Lawrence C. Introduction to cyclotomic fields, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 83 (1997), pp. xiv+487 | MR 1421575 | Zbl 0484.12001